Path: ...!local-3.nntp.ord.giganews.com!local-4.nntp.ord.giganews.com!Xl.tags.giganews.com!local-2.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail NNTP-Posting-Date: Tue, 31 Dec 2024 20:04:49 +0000 Subject: Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary, effectively) Newsgroups: sci.math References: <45a632ed-26cc-4730-a8dd-1e504d6df549@att.net> <3d2fe306aa299bc78e94c14dadd21645d8db9829@i2pn2.org> <2163aa0c0efba66c813e8ebda5ef5ece6d19ea34@i2pn2.org> From: Ross Finlayson Date: Tue, 31 Dec 2024 12:04:51 -0800 User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101 Thunderbird/38.6.0 MIME-Version: 1.0 In-Reply-To: Content-Type: text/plain; charset=utf-8; format=flowed Content-Transfer-Encoding: 8bit Message-ID: Lines: 57 X-Usenet-Provider: http://www.giganews.com X-Trace: sv3-qLqtdG31MKAnLw6KUzNispP7vRQ8QKwz/DZBD7iemhZBQkib8rKatgWQIMm/rkA61lypxqqw0SRQTts!IwMS54hXvwP2SQFGTApPR+OioVn+blHublRZYFRGjMISwdYXvaIGi1G+BxmUHwn9I1QTDIWrtRQ= X-Complaints-To: abuse@giganews.com X-DMCA-Notifications: http://www.giganews.com/info/dmca.html X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly X-Postfilter: 1.3.40 Bytes: 4111 On 12/31/2024 09:17 AM, FromTheRafters wrote: > Richard Damon was thinking very hard : >> On 12/30/24 4:38 PM, WM wrote: >>> On 30.12.2024 17:08, Richard Damon wrote: >>>> On 12/30/24 3:44 AM, WM wrote: >>>>> On 30.12.2024 01:36, Richard Damon wrote: >>>>>> On 12/29/24 5:31 PM, WM wrote: >>>>> >>>>>>> My theorem: Every union of FISONs which stay below a certain >>>>>>> threshold stays below that threshold. >>>>>>> Find a counterexample. Don't claim it but prove it. Fail. >>>>> >>>>>> But what does that prove? >>>>> >>>>> That proves the existence of dark numbers. >>>> >>>> How? >>>> >>>> It shows that there are numbers you didn't look at, but you admit to >>>> not looking at all the numbers. >>> >>> Neither can you look at more numbers. >> >> Of course I can look at "more" numbers, as I can look at the one after >> where you stopped. >> >>>> >>>>>> The finite numbers are finite, and that is it. >>>>> >>>>> FISONs do not grow by unioning them. All FISONs and their unions >>>>> stay below 1 % of ℕ. >>> >>>>> Proof: Every FISON that is multiplied by 100 remains a FISON that >>>>> can be multiplied by 100 without changing this property. >>>>> >>>>> If ℕ is an actually infinite set then it is not made by FISONs. >>> >>> That is what I have been telling you for a long time. >> >> But there is no Natural Number that isn't in a FISON, which is NOT >> what you have been saying. >> >> You don't seem to understand the difference between a set and its >> members. > > He has in the past explicitly stated such. Something like 'a set is > nothing more than its elements' -- I don't think he has changed his tune. Sets are defined by their elements, then though all the relations of sets are defined by all the relations of sets, in set theory, a theory with one relation, elt. Then there's class/set distinction, that's another fine point ignored by all inductive bots.